I am using LMDE (Linux Mint Debian Edition).
By default, this ships with a quite old Firefox version (21.0). And if you install Iceweasel, it does not get better.

There are a number of forum posts about installing a newer version of Iceweasel OR Firefox. Ultimately a lot of them lead here: http://mozilla.debian.net/, where I am asked to add this repository to /etc/apt/sources.list

deb http://cdn.debian.net/debian experimental main

and to install as follows

$ apt-get update
$ apt-get install -t experimental iceweasel

Unfortunately, this gets me into the following error:

The following packages have unmet dependencies:
iceweasel : Depends: libc6 (>= 2.17) but 2.13-37 is to be installed
Depends: xulrunner-23.0 (>= 23.0-2) but it is not going to be installed
E: Unable to correct problems, you have held broken packages.

Unfortunately, the only other option that I'm aware of (besides waiting) is to repackage the .deb package from the firefox-next PPA and change the dependencies listed. You would have to do that for each version though (and hope the only difference is the ~ubuntu appended to the version number rather than compiled against a different version)

If a real rolling release is important to you and you're comfortable with CLI then I'd highly recommend taking the leap to ArchLinux or Gentoo.